Parasitic Motions of 3-PRS Parallel Mechanisms with Two Different Branch Chain Arrangements

Abstract

The kinematics of 3-PRS parallel mechanisms (PMs), an important category of 2R1T PMs, is very important for the motion control and error compensation of PMs. In order to produce a spindle head that could meet the requirements of various machining accuracy levels, a definite kinematics analysis of the PMs par should be taken with caution firstly. Here, the mathematical models of two 3-PRS PMs with different branch chain arrangements were established using the Euler angle. The kinematic equations of two 3-PRS PMs were obtained by establishing the closed-vector loop equation according to structural conditions. Parasitic motions that would cause a great effect were obtained, and their variations with structural and kinematic parameters were also obtained through MATLAB. Since the relationship between parasitic motions, and the given structural and kinematic parameters, is inextricable, an analysis is performed for the purpose of reducing or eliminating parasitic motion to ensure the accuracy of end-effector movement. The caused effect of parasitic motion can be improved with scale optimization and precision compensation. Lastly, two improved 3-PRS PMs without the parasitic motion of different branch chain arrangements used as the spindle head were obtained. Parasitic motion was related to the mechanism configuration and could be avoided in the configuration design stage with the comprehensive optimization of the structural and kinematic parameters, which provides a reference to obtain a sufficient workspace and improve the precision of traditional five-axis CNC machine tools.

1. Introduction

In recent decades, PMs have attracted much research focus. Since most practical applications do not require six degrees of freedom (DOFs), research on lower-mobility PMs with a high speed, stiffness, and load-carrying capacity is gradually expanding into many fields [1]. Lower-mobility PMs, whose DOFs are fewer than six, are usually implemented in industrial applications, such as machining [2,3]. Two rotational DOFs and one translation DOF (2R1T) PMs are simple to design and have low fabrication, actuation, control, and maintenance costs. In industrial fields, 2R1T PMs, as the spindle head of five-DOF computer numerical control (CNC) machine tools, require a high speed and high precision for machining. The sprint Z3 head [4], Tricept hybrid robot [5], and Exechon parallel robot [6,7] are typical examples of successful 2R1T PMs. As a key component that greatly determines the function and performance of robots or CNC machine tools, such as spindle heads and high-performance reducers (such as RV reducers), 2R1T PMs are an effective solution to related problems.

The most famous 2R1T PM may be the 3-RPS PM proposed by Hunt [8], where R denotes a revolute pair, P a prismatic pair, and S a spherical joint. The 3-RPS PM or its variant, 3-PRS PM, has attracted much attention, and has broad application prospects and potential in industrial fields [9,10]. Research was performed regarding kinematics analysis [11,12], dimensional synthesis [13,14,15], singularity [16], and dynamics [17,18].

Carretero et al. [19] proposed the concept of the parasitic motion of the 3-PRS PM, which is unwanted in the mechanism but has a great effect on the application of the mechanism. Herrero et al. [20] discussed whether the accompanying motion would affect the dynamic characteristics of the mechanism and suggested that the parasitic motion depends on the used transformation matrix combination. Mathematical methods, such as fractional differential equations, have risen to prominence in scientific and engineering fields [21,22,23,24]. Gallardo [25] used the Newton homotopy continuation method to analyze the kinematics of a 3-RRS PM, facilitating calculating the accessible workspace.

Yang et al. [26] analyzed lightweight 3-DOF flexural prismatic joints based on 2R1T 3-PRS PM. Liu et al. [27] proposed a type of PPS PM to simplify the description of the parasitic motion of 3-PRS PM, satisfying the specific geometric condition that three planes of branch chains intersect at a common line. Li et al. [28] systematically described the relations between the parasitic motion and the chain arrangements of a 3–-PRS PM and synthesized a series of asymmetric PMs without parasitic motion. Xu et al. [29] proposed a developed overconstrained 2UPR–RPU–RPR PM that improved the bearing capacity of the mechanism. Hu et al. [30] developed an optimal selection that considered both transmission and constraint performance on the basis of the equivalent mechanism of the asymmetric 3-PRS mechanism.

Ranjan et al. [31] changed the design parameters to minimize the parasitic motion through the size ratio of the moving and fixed platform of the reconfigurable 3–RPS parallel robot. Fang [32] used interval analysis to simplify the complex constrained optimization problem for the optimal design of 3–-PRS parallel manipulators in order to minimize parasitic motions.

Chen et al. [33] deduced the synthesis criteria and geometric conditions of a 1T2R PM without parasitic motion, which were obtained by analyzing the condition of the rotational axes of a PM without parasitic motion on the basis of the spatial geometry theory and exploring the spatial position relationship between the plane of the moving platform and the constraint wrenches using the screw theory. The positions of the constraint wrenches and rotational axes directly caused parasitic motions that were verified by kinematics analysis. On the basis of the inverse kinematics of the 3-PRS PM, Hao et al. [34] established the workspace of a 3–-PRS PM through the stroke limit of the sliding rod and the rotating angle constraint condition of the spherical pair and the revolute pair. Xia et al. [35] analyzed a UPS–UPR–S PM with two degrees of freedom and concluded that the PM has no parasitic motion.

As an inherent kinematic feature of a PM, parasitic motion leads to complex kinematics, requires real-time compensation, and hinders calibration. For applied 2R1T PMs as the spindle head of CNC machine tools, it is greatly practically significant to obtain 2R1T PMs with an accurate attitude adjustment ability and a large workspace through kinematic analysis. Many scholars have studied the minimization and elimination of the parasitic motions of 2R1T PMs. The performance of 2R1T PMs is closely related to its two shafts, and parasitic motions may affect the accuracy of end effectors for 2R1T PMs. Since there are countless kinds of 3-PRS PMs under different structural and branched-chain arrangements, it is necessary to study the inherent relation between the parasitic motions and configuration of 2R1T PMs. Parasitic motion varies with the configuration of the PM.

On the basis of the relative relationship between the three branch chain planes and the three centers of the spherical joints, this paper analyzes the kinematics of two 3-PRS 2R1T PMs that had different structures and the same branch chains under specific circumstances. Therefore, for the PM with precision requirements, such as machine tool processing, parasitic motion must be analyzed. If a parasitic motion exists and cannot be ignored, topological optimization or precision compensation is needed to eliminate its influence and render the end-effector motion more accurate. Last, two improved 3-PRS PMs without the parasitic motion of different branch chain arrangements were established, which provides a reference for 2R1T PMs used as the spindle head of five-DOF CNC machine tools to provide a high speed and high precision in machining in industrial fields. The remainder of this paper is organized as follows. Section 2 first establishes the mathematical model of the 3-PRS PM I and the parasitic motions by establishing a closed-vector loop equation. Second, according to the parasitic motion equation, the influence factors of the parasitic motion are analyzed. Lastly, an improved 3-PRS PM I without parasitic motion is presented via the comprehensive optimization of structural and kinematic parameters. In Section 3, the mathematical model and parasitic motions of the 3-PRS PM II are derived, and the influence factors of parasitic motion and the improved 3-PRS PM II are analyzed. Conclusions are drawn in Section 4.

2. 3-PRS PM I

2.1. 3-PRS PM I Modeling

The 3-PRS PM includes a moving platform S₁S₂S₃, a fixed platform B₁B₂B₃, and three PRS branch chains. The prismatic joint is actuated by a linear actuator. The axis of the revolute joint R_i (i = 1, 2, 3) is perpendicular to the axis of the prismatic joint P_i. To facilitate the design and analysis of 3-PRS PMs, the fixed platform B₁B₂B₃ is set as an equilateral triangle with circumradius R and a central point O_B, and the moving platform S₁S₂S₃ is also an equilateral triangle with circumradius r and a central point O_m. The fixed platform and the moving platform are connected by three PRS branch chains. The spherical pair S_i is evenly distributed 120° on the moving platform S₁S₂S₃. This 3-PRS PM is called a 3-PRS PM I in this paper. The model is shown in Figure 1.

As shown in Figure 2, the fixed coordinate system O_B-XYZ ({O_B} for short) is connected to the origin O_B of the fixed platform B₁B₂B₃. The direction of the Y axis is from O_B to B₃. The direction of the X axis is parallel to B₁B₂. The Z axis is perpendicular to the plane B₁B₂B₃ and the direction is upward.

The moving coordinate system O_m-xyz ({O_m} for short) is connected to the origin O_m of the moving platform S₁S₂S₃. The direction of the y axis is from O_m to S₃. The direction of the x axis is parallel to S₁S₂. The z axis is perpendicular to the plane S₁S₂S₃ and the direction is upward.

2.2. Kinematics Analysis

The aim of kinematics analysis is to obtain the constraint equations according to the geometric characteristics of the PMs. In this paper, a y-x-z Euler angle is introduced to describe the posture of the moving platform in the {O_B} coordinate system; that is, let the rotation angle around the y axis be first θ, then the rotation angle around the x axis can be ψ, and, finally, the rotation angle around the z axis can be ϕ. The rotation matrix T is:

$$\begin{array}{l}\mathit{T}={\mathit{T}}_{yxz}={\mathit{R}}_y(\theta ){\mathit{R}}_x(\psi ){\mathit{R}}_z(\phi )\\ =\left[\begin{array}{ccc}\mathrm{c}\theta \mathrm{c}\phi +\mathrm{s}\psi \mathrm{s}\theta \mathrm{s}\phi & -\mathrm{c}\theta \mathrm{s}\phi +\mathrm{s}\psi \mathrm{s}\theta \mathrm{c}\phi & \mathrm{c}\psi \mathrm{s}\theta \\ \mathrm{c}\psi \mathrm{s}\phi & \mathrm{c}\psi \mathrm{c}\phi & -\mathrm{s}\psi \\ -\mathrm{s}\theta \mathrm{c}\phi +\mathrm{s}\psi \mathrm{s}\theta \mathrm{s}\phi & \mathrm{s}\theta \mathrm{s}\phi +\mathrm{s}\psi \mathrm{c}\theta \mathrm{c}\phi & \mathrm{c}\psi \mathrm{c}\theta \end{array}\right]=\left[\begin{array}{ccc}{\mathrm{T}}_{11}& {\mathrm{T}}_{12}& {\mathrm{T}}_{13}\\ {\mathrm{T}}_{21}& {\mathrm{T}}_{22}& {\mathrm{T}}_{23}\\ {\mathrm{T}}_{31}& {\mathrm{T}}_{32}& {\mathrm{T}}_{33}\end{array}\right]\end{array}$$

(1)

where c represents the cosine and s represents the sine.

$${\mathit{R}}_y(\theta )=\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right]{\mathit{R}}_x(\psi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \psi & -\sin \psi \\ 0& \sin \psi & \cos \psi \end{array}\right]{\mathit{R}}_z(\varphi )=\left[\begin{array}{ccc}\cos \phi & \sin \phi & 0\\ -\sin \phi & \cos \phi & 0\\ 0& 0& 1\end{array}\right]$$

According to Figure 2 and Figure 3, the vectors of the joint position and motion direction are obtained, as shown in

Table 1. Joint position vectors and motion direction vectors.

Expression	Physical Meaning (i = 1, 2, 3)
$p={\left[\begin{array}{ccc}x& y& z\end{array}\right]}^{\mathrm{T}}$	In the {OB} coordinate system, the vector from the origin OB to the origin Om.
$a_i={\left[\begin{array}{ccc}{a}_{ix}& a_{iy}& a_{iz}\end{array}\right]}^{\mathrm{T}}$	In the {OB} coordinate system, the vector from the origin OB to point Bi.
$l_i={\left[\begin{array}{ccc}{l}_{ix}& l_{iy}& l_{iz}\end{array}\right]}^{\mathrm{T}}$	In the {OB} coordinate system, the vector from the origin OB to the spherical joint center Si.
${\mathit{S}}_i={\left[\begin{array}{ccc}{S}_{\mathrm{ix}}& S_{iy}& S_{iz}\end{array}\right]}^{\mathrm{T}}$	In the {OB} coordinate system, the vector from the origin Om to the spherical joint center Si.
${\mathit{S}\prime }_i^{}={\left[\begin{array}{ccc}{S^{\prime }}_{ix}^{}& {S^{\prime }}_{iy}^{}& {S^{\prime }}_{iz}^{}\end{array}\right]}^{\mathrm{T}}$	In the {Om} coordinate system, the vector from the origin Om to the spherical joint center Si.
${\mathit{d}}_{\mathit{i}}$	The moving distance of the prismatic pair Pi

.

In the {O_B} coordinate system, the vectors of the kinematic pairs of the three branch chains are as follows.

The position vectors of B₁, B₂, and B₃ in the {O_B} coordinate system are a₁, a₂, and a₃:

$$\begin{array}{l}{\mathit{a}}_1={\left[\begin{array}{ccc}-\frac{\sqrt{3}R}{2}& -\frac{R}{2}& 0\end{array}\right]}^{\mathrm{T}}\\ {\mathit{a}}_2={\left[\begin{array}{ccc}\frac{\sqrt{3}R}{2}& -\frac{R}{2}& 0\end{array}\right]}^{\mathrm{T}}\\ {\mathit{a}}_3={\left[\begin{array}{ccc}0& R& 0\end{array}\right]}^{\mathrm{T}}\end{array}$$

(2)

The position vectors of the prismatic joint P_i (i = 1, 2, 3) in the {O_B} coordinate system are:

$$\begin{array}{l}{\mathit{P}}_1={\left[\begin{array}{ccc}-\frac{\sqrt{3}R}{2}& -\frac{R}{2}& d_1\end{array}\right]}^{\mathrm{T}}\\ {\mathit{P}}_2={\left[\begin{array}{ccc}\frac{\sqrt{3}R}{2}& -\frac{R}{2}& d_2\end{array}\right]}^{\mathrm{T}}\\ {\mathit{P}}_3={\left[\begin{array}{ccc}0& R& d_3\end{array}\right]}^{\mathrm{T}}\end{array}$$

(3)

The position vectors of the spherical pairs S₁, S₂, and S₃ in the {O_m} coordinate system are S₁′,S₂′, and S₃′:

$$\begin{array}{l}{\mathit{S}}_1^{\prime }={\left[\begin{array}{ccc}-\frac{\sqrt{3}\mathrm{r}}{2}& -\frac{r}{2}& 0\end{array}\right]}^{\mathrm{T}}\\ {\mathit{S}}_2^{\prime }={\left[\begin{array}{ccc}\frac{\sqrt{3}r}{2}& -\frac{r}{2}& 0\end{array}\right]}^{\mathrm{T}}\\ {\mathit{S}}_3^{\prime }={\left[\begin{array}{ccc}0& r& 0\end{array}\right]}^{\mathrm{T}}\end{array}$$

(4)

The position vectors of the spherical pairs S₁, S₂, and S₃ are represented as l_i (i = 1, 2, 3) in the {O_B} coordinate system by S₁′,S₂′, and S₃′ using the closed-loop vector method.

$$l_{\mathit{i}}=S_{\mathit{i}}+p=T\cdot {S^{\prime }}_{\mathit{i}}+p$$

(5)

Substituting all the vectors aforementioned into Equation (5) yields:

$$\left\{\begin{array}{c}{l}_{1x}=x+T_{11}{b}_{1x}^{\prime }+T_{12}{b}_{1y}^{\prime }\\ l_{1y}=y+T_{21}{b}_{1x}^{\prime }+T_{22}{b}_{1y}^{\prime }\end{array}\right.$$

(6a)

$$\left\{\begin{array}{c}{l}_{2x}=\\ l_{2y}=\end{array}\right.\begin{array}{c}x+T_{11}{b}_{2x}^{\prime }+T_{12}{b}_{2y}^{\prime }\\ y+T_{21}{b}_{2x}^{\prime }+T_{22}{b}_{2y}^{\prime }\end{array}$$

(6b)

$$\left\{\begin{array}{c}{l}_{3x}=x+T_{12}{b}_{3y}^{\prime }\\ l_{3y}=y+T_{22}{b}_{3y}^{\prime }\end{array}\right.$$

(6c)

According to the constraints of the revolute pairs of the mechanism’s structure, each branch chain of the 3-PRS PM I only moves in the corresponding plane in Figure 4. Constrained by the structural conditions of the mechanism, each branch chain can obtain three constraint equations through Equation (6a–c) as follows:

$$l_{1\mathrm{y}}=\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.l_{1\mathrm{x}}$$

(7a)

$$l_{2\mathrm{y}}=-\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.l_{2\mathrm{x}}$$

(7b)

$$l_{3\mathrm{x}}=0$$

(7c)

The solution of the vector p is obtained by combining Equations (6) and (7):

$$x=r\left(\cos \theta \sin \phi -\sin \psi \sin \theta \cos \phi \right)$$

(8)

$$y=\frac{r}{2}(\sqrt{3}\cos \psi \sin \phi +\cos \psi \cos \phi +\sqrt{3}\cos \theta \sin \phi -\sqrt{3}\sin \psi \sin \theta \cos \phi -\cos \theta \cos \phi -\sin \psi \sin \theta \sin \phi )$$

(9)

$$\phi =\arctan \frac{\sin \psi \sin \theta }{\cos \psi +\cos \theta }$$

(10)

According to Equations (8)–(10), it is known that the moving platform S₁S₂S₃ produces one rotation ϕ and two translation movements, x and y, of the moving platform center point O_m along the x and y axes.

The small amplitude motions of the constrained degrees of freedom are called parasitic motions, that is the non-independent degrees of freedom, which are obtained by the constraint analysis of the center point of the moving platform. The existence of parasitic motion will cause the center point of the mechanism moving platform move in a non-direction of the degree of freedom, which brings great difficulties to the dimensional synthesis and trajectory planning of the mechanism. It will cause an effect on the precision of the mechanism and is harmful to the mechanism. The minimization and elimination of the parasitic motion have a significant influence on the practical application. Additionally, the motion parameter ϕ is related to the kinematic parameters ψ and θ. The motion parameters x and y are related to the structural parameter r and the kinematic parameters ψ and θ.

2.3. Parasitic Motion Analysis

The structural parameters of the 3-PRS PM I in Figure 1 are given in

Table 2. Structural parameters of 3-PRS PM I.

Parameter	Value
Equivalent circumradius of the fixed platform B1B2B3, R [mm]	125
Equivalent circumradius of the moving platform S1S2S3, r [mm]	80
Range of the moving distance of the pairs Pi, di [mm]	[105, 195]
Ranges of the kinematic parameters ψ and θ	[−60°, 60°]

. The ranges of the kinematic parameters ψ and θ are [−60°, 60°], which fully satisfies the rotation angles required by the PM.

When the structural parameter r of the moving platform S₁S₂S₃ is given as 250 mm, 175 mm, and 100 mm and the ranges of the kinematic parameters ψ and θ are all as [−60°, 60°], the parasitic motions x and y can be obtained through MATLAB, as shown in Figure 5 and Figure 6.

The CPU running times of the parasitic motions x with the moving platform m given as 250 mm, 175 mm, and 100 mm through MATLAB are 2.3400 s, 1.9968 s, and 1.9032 s, while the CPU running times of the parasitic motions y are 1.4820 s, 1.4976 s, and 1.5600 s.

When the circumradius r of the moving platform S₁S₂S₃ is given as 250 mm, the kinematic parameters ψ and θ are given as from [−60°, 60°] to [−5°, 5°]. The motion parameter ϕ obtained through MATLAB is shown as in Figure 7.

The CPU running times of the motion parameter ϕ with kinematic parameters ψ and θ given as [−60°, 60°], [−30° 30°], and [−5°, 5°] through MATLAB are 1.9032 s, 0.4524 s, and 0.0936 s.

According to Figure 5 and Figure 6, the parasitic motions x and y are symmetrically distributed and the value of x and y at ψ and θ at [0, 0] is 0, that is, there is no parasitic motion occurring.

When given the ranges of the kinematic parameters ψ and θ, the maximum values of the parasitic motions x and y with the circumradius r = 250 mm and r = 100 mm of the moving platform S₁S₂S₃ are obtained as outlined in

Table 3. Variations in the parasitic motions x and y with the kinematic parameters ψ and θ.

Angles	x-Max [mm] (r = 250)	x-Max [mm] (r = 100)	y-Max [mm] (r = 250)	y-Max [mm] (r = 100)
ψ and θ ∈ [−60°, 60°]	80.6789	32.2716	62.5000	25
ψ and θ ∈ [−45°, 45°]	58.9256	23.5702	36.6117	14.6447
ψ and θ ∈ [−30°, 30°]	30.9295	12.3718	16.7468	6.6987
ψ and θ ∈ [−15°, 15°]	8.3684	3.3474	4.2593	1.7037
ψ and θ ∈ [−10°, 10°]	3.7688	1.5075	1.8990	0.7596
ψ and θ ∈ [−5°, 5°]	0.9495	0.3798	0.4757	0.1903

.

When given the ranges of the structural parameters r of the moving platform S₁S₂S₃ as [35, 350] mm, the values of the parasitic motions x with the given ψ and θ has a linear relationship with the circumradius r, as shown in Figure 8.

When given the ranges of the structural parameters r of the moving platform S₁S₂S₃ as [35, 350] mm, the values of the parasitic motions y with the given ψ and θ also have a linear relationship with the circumradius r, as shown in Figure 9.

2.4. Improved 3-PRS PM I without Parasitic Motion

As can be seen from Figure 5 and Figure 6, as the ranges of the kinematic parameters ψ and θ increase, the parasitic motions become more obvious. That is to say, when the structural parameters remain unchanged, the larger the workspace of the mechanism, the more obvious the parasitic motion is. When the kinematic parameters are in the smaller angle range, for instance, [−5°, 5°], the parasitic motions along with the movement are not obvious and can be neglected. However, when the amplitude variation in the kinematic parameters is bigger, the amplitude variation in the parasitic motions change and is also larger. That is, the parasitic motions are increasingly obvious, and the influence on the mechanism’s performance may be also increasing. Specifically, the parasitic motions reach the maximum when the PM moves to its limit position, which causes operation error and affects the running accuracy of the PM. Thus, in order to reduce or eliminate parasitic motions, it is important to optimize the design of the PM and kinematic parameters for the PMs used for large piece machining.

In order to avoid the parasitic motions of the 3-PRS PM I described in Section 2.1, Equations (8)–(10) are equal to 0 and are solved for the structural parameter r and the kinematic parameters ψ and θ; the value of the structural parameter r is 0. The improved 3-PRS PM I without parasitic motion is shown in Figure 10.

When given the structural parameters r and the kinematic parameters of the improved 3-PRS PM I, the displacement component z, the velocity, and acceleration of the moving platform S₁S₂S₃ are solved as shown in Figure 11.

3. 3-PRS PM II

3.1. 3-PRS PM II Modeling

Set the fixed platform B₁B₂B₃ and the moving platform S₁S₂S₃ of 3-PRS PM as isosceles triangles. This 3-PRS PM is called as 3-PRS PM II in this paper and the model is shown in Figure 12.

The length of side O_BB₃ is b and the length of sides B₁B₃ and B₂B₃ is a, while the length of side O_mB₃ is d and the length of the two sides b₁b₃ and b₂b₃ is c. Three PRS branch chains connect the fixed platform B₁B₂B₃ and the moving platform S₁S₂S₃. The mechanism diagram of the 3-PRS PM I is shown in Figure 13.

As shown in Figure 12, the fixed coordinate system O_B-XYZ ({O_B} for short) is connected to the origin O_B of the fixed platform B₁B₂B₃. The direction of the Y axis is from O_B to B₃. The direction of the X axis is from O_B to B₂. The Z axis is perpendicular to the fixed platform B₁B₂B₃ and the direction is upward.

The moving coordinate system O_m-xyz ({O_m} for short) is connected to the origin O_m of the moving platform S₁S₂S₃. The direction of the y axis is from O_m to S₃. The direction of the x axis is from O_m to S₂. The z axis is perpendicular to the plane S₁S₂S₃ and the direction is upward.

3.2. Kinematics Analysis

The joint position vectors and motion direction vectors are obtained, as shown in Table 1. According to Figure 13 and Figure 14, the vectors of the kinematic pairs of the three branches in the {O_B} coordinate system are as follows.

The position vectors of B₁, B₂, and B₃ in the {O_B} coordinate system are a₁, a₂, and a₃:

$$\begin{array}{l}{\mathit{a}}_1={\left[\begin{array}{ccc}-a& 0& 0\end{array}\right]}^T\\ {\mathit{a}}_2={\left[\begin{array}{ccc}a& 0& 0\end{array}\right]}^T\\ {\mathit{a}}_3={\left[\begin{array}{ccc}0& b& 0\end{array}\right]}^T\end{array}$$

(11)

The position vectors of the prismatic joint P_i (i = 1, 2, 3) in the {O_B} coordinate system are:

$$\begin{array}{l}{\mathit{P}}_1={\left[\begin{array}{ccc}-a& 0& d_1\end{array}\right]}^{\mathrm{T}}\\ {\mathit{P}}_2={\left[\begin{array}{ccc}a& 0& d_2\end{array}\right]}^{\mathrm{T}}\\ {\mathit{P}}_3={\left[\begin{array}{ccc}0& b& d_3\end{array}\right]}^{\mathrm{T}}\end{array}$$

(12)

The position vectors of the spherical pairs S₁, S₂, and S₃ in the {O_m} coordinate system are S₁′, S₂′, and S₃′:

$$\begin{array}{l}{\mathit{S}\prime }_1^{}={\left[\begin{array}{ccc}-c& 0& 0\end{array}\right]}^{\mathrm{T}}\\ {\mathit{S}\prime }_2^{}={\left[\begin{array}{ccc}c& 0& 0\end{array}\right]}^{\mathrm{T}}\\ {\mathit{S}\prime }_3^{}={\left[\begin{array}{ccc}0& e& 0\end{array}\right]}^{\mathrm{T}}\end{array}$$

(13)

The position vectors of spherical pairs S₁, S₂, and S₃ are represented as l_i (i = 1,2,3) in the {O_B} coordinate system {B} by S₁′, S₂′, and S₃′ using a closed loop vector method:

$$l_{\mathbf{i}}=S_{\mathit{i}}+p=T\cdot {S^{\prime }}_{\mathit{i}}^{}+p$$

(14)

Substituting all the vectors aforementioned into Equation (14) yields:

$$\left\{\begin{array}{c}{l}_{1x}=x+T_{11}{S}_{1x}^{\prime }\\ l_{1y}=y+T_{21}{S}_{1x}^{\prime }\end{array}\right.$$

(15a)

$$\left\{\begin{array}{c}{l}_{2x}=\\ l_{2y}=\end{array}\right.\begin{array}{c}x+T_{11}{S}_{2x}^{\prime }\\ y+T_{21}{S}_{2x}^{\prime }\end{array}$$

(15b)

$$\left\{\begin{array}{c}{l}_{3x}=x+T_{12}{S}_{3y}^{\prime }\\ l_{3y}=y+T_{22}{S}_{3y}^{\prime }\end{array}\right.$$

(15c)

According to the constraints of the revolute pairs of the mechanism structure, each branch chain of the PM only moves in the corresponding plane as in Figure 15. Constrained by the structural conditions of the mechanism, each branch chain can obtain three constraint equations through Equations (15a)–(15c), as follows:

$$l_{1\mathrm{y}}=0$$

(16a)

$$l_{2\mathrm{y}}=0$$

(16b)

$$l_{3\mathrm{x}}=0$$

(16c)

The solution of vector p is obtained by combining Equation (16):

$$x=-e\sin \psi \sin \theta ,y=0, \varphi =0$$

(17)

According to Equation (17), it is known that the moving platform S₁S₂S₃ produces one translation movement x of the moving platform center point along the axis x. The parasitic motion x is related to the structural parameter e and kinematic parameters ψ and θ. When the kinematic parameters ψ and θ and structural parameter e of the moving platform S₁S₂S₃ are given, the parasitic motions x can be obtained.

3.3. Parasitic Motion Analysis

The model shown in Figure 12 was established in SolidWorks, and the structural parameters are given in

Table 4. Structural parameters of 3-PRS PM II.

Parameter	Value
Length of side OBS2 of the fixed platform B1B2B3, a [mm]	125
Length of side OBB3 of the fixed platform B1B2B3, b [mm]	150
Length of side OmS2 of the moving platform S1S2S3, c [mm]	80
Length of side OmS3 of the moving platform S1S2S3, e [mm]	60
Range of the equivalent link length of the pairs Pi, di [mm]	[105, 195]
Ranges of the kinematic parameters ψ and θ	[−60°, 60°]

.

The ranges of the kinematic parameters ψ and θ are also [−60°, 60°], which fully satisfies the rotation angles required by the PM.

When the structural parameters e of side O_mS₃ of the moving platform S₁S₂S₃ are given as 250 mm, 175 mm, and 100 mm and the ranges of the kinematic parameters ψ and θ are given as [−60°, 60°], the parasitic motion x can be obtained through MATLAB, as shown in Figure 16.

The CPU running times of parasitic motions x with the length of side O_mS₃ of the moving platform S₁S₂S₃ given as 250 mm, 175 mm, and 100 mm and obtained through MATLAB are 2.1060 s, 1.8564 s, and 1.6848 s.

When given the ranges of the kinematic parameters ψ and θ, the maximum values of the parasitic motions x with the circumradius e = 250 mm and e = 100 mm of the moving platform S₁S₂S₃ are obtained as outlined in

Table 5. Variations in parasitic motion x with kinematic parameters ψ and θ.

Angles ψ and θ	x-Max [mm] (e = 250)	x-Max [mm] (e = 100)
ψ and θ ∈ [−60°, 60°]	187.5	75
ψ and θ ∈ [−45°, 45°]	125	50
ψ and θ ∈ [−30°, 30°]	62.5	25
ψ and θ ∈ [−15°, 15°]	16.75	6.6987
ψ and θ ∈ [−10°, 10°]	7.5384	3.0154
ψ and θ ∈ [−5°, 5°]	1.8990	0.7596

.

When the ranges of the structural parameters e of the moving platform S₁S₂S₃ are given as [35, 350] mm, the values of the parasitic motions x with the given ψ and θ parameters have a linear relationship with the circumradius e, as shown in Figure 17.

3.4. Improved 3-PRS PM II without Parasitic Motion

In order to avoid the parasitic motions of the 3-PRS PM II described in Section 3.1, Equation (17) is equal to 0 and solved for the structural parameter e, and for the kinematic parameters ψ and θ, the value of the structural parameter e is 0. The improved 3-PRS PM II without parasitic motion is shown in Figure 18.

When given the structural parameters e and the kinematic parameters of the improved 3-PRS PM II, the displacement component Z, the velocity, and acceleration of the moving platform S₁S₂S₃ are solved as shown in Figure 19.

4. Conclusions

The general 2R1T PMs not only produce motions in the direction of DOF, but they also produce parasitic motion which incomplete motion constrained in the directions of other DOFs, which results in all points of the moving platform moving in a specific direction. As an inherent kinematic feature of PMs, parasitic motion leads to complex kinematics, requires real-time compensation, and hinders calibration. To study the inherent relation between the parasitic motion and the structure of 3-PRS PMs, two improved 3-PRS PMs of different branch chain arrangements are proposed in this paper.

(1) The parasitic motions are obtained using the closed-loop vector method according to the constraint conditions. According to Section 2.3 and Section 3.3, it is known that parasitic motions are inherent kinematic features and vary with the configurations of the two mathematical models of 3-PRS PM I and II.

(2) The obtained parasitic motions are related to the kinematic parameters and structural parameters of 3-PRS PM through analysis. When the kinematic parameters vary in a relatively small range, for instance, [−5°, 5°], the parasitic motions are not obvious and can be ignored. When the kinematic parameters change greatly, the amplitude is relatively large and should be taken into account.

(3) Given the structural and kinematic parameters, the maximum and minimum of parasitic motions in the limit position are pointed out and two improved 3-PRS PMs without parasitic motions are obtained. According to Section 2.4 and Section 3.4, parasitic motion can be avoided in the configuration at the design stage.

The results show that parasitic motions have an influence on kinematic analysis, which is meaningful to traditional five-axis CNC machine tools while machining the parts of the complex structure. For the 2R1T PMs with mixed degrees of freedom, studying and clarifying the trajectory of motions is of great practical significance to the motion control, error compensation, and calibration of PMs.